'·"··' ·. CHAPTER- 6 Some Astronomical Principles : An Indian Perspective Two types of knowledge: apariividyli (inferior knowledge) and parlividyli (superior or spiritual knowledge) have been accepted in Indian tradition. An act of worship done with a specific worldly desire was .consid~red an inferior form of worship and was popular with the kings. Aparlividya enables man to attain material progress, enrichment and fulfilment of life and parlividyli ensures attainment of self-realization or salvation in life (Chand. Up. 7.1.7; MuYJc!a. Up. 1.2.4-5). The Vedic people, in general, made interesting synthesis by adopting nitya (perpetual or daily) and klimya (optional for wisl¥ fulfilment) sacrifices or offerings. The first was supposed to .bring happiness to the family and second was for material progress. The perpetual daily sacred fires and the optional fires were placed on altars of various shapes. As to the reasons, which might have induced the ancient Indians to devise all these strange shapes, the J?,g-Veda (1.15.12) says, 'He who desires heaven, may construct falcon shaped altar, for falcon is the best flyer among the birds'. These may appear to be superstitious fancies but led to important contributions in geometry and mathematics because of their conviction in · social value systems. To find the right time for religious, agricultural, new.year and other social festivals gave the motivation for recording of recurrence of repeated events from seasons, stars, movement of planets, moon etc. This helped to develop many a framework for mapping of movements of heavenly bodies with reference ·to East, West, 'North and South points, nak:jatras, calendar, yuga, inahliyuga and movement of planets for mean and true positions of planets. Various mathematical 1and trigonometrical tables were also formulated for better and better results. The Vedlinga Jyoti~a 1 mentions as under: 102 Having saluted Time with bent head, as also Goddess Saraswati, I shall explain the lore of Time, as enunciated by sage Lagadha. As the crests on the head of the peacocks, the jewels on the serpents, so is the (jyoti~a ga1]itam) held at the head of alllores among all vedliY!ga slistras. G~ita is a variant reading for jyoti~a meaning computation, which is the essence of this science. Another tradition, which has enriched specialized activities in mathematics and astronomy is the guru-si~ya-paramparli (teacher-student tradition). Different recensions of Vedic schools, sulbakaras, jyoti~karas (Varahamihira names tWenty scholar before him), Kusumpura school, schools of Ujjain and Asmakadesa, and Jain and Buddhist schools are also well-known in this coimection. Beside these, there were commercial and ot9er problems, which were tackled for­ lokavyavahlirlirtha, used for 'common people. The restriction and emphasis were also assured on social use and value systems, which helped people to take up different activities for commerce, education and other areas. These helped undoubtedly to concretise knowledge resulting in original contributions to mathematics and astronomy. Construction of altars and nature of knowledge : The ritual connection of Indian geometry, as elaborated by Thibaut and Burk, has been intensively discussed by Datta, Seidenberg, Sen and Bag, ~d others.2 The ceremonies were performed on the top of altars built either in the sacrificer's house or on a nearby plot of ground. The altar is a specified raised area, generally made of bricks for keeping the fire. The fire altars were of two types. The perpetual fires (nitya agni) were constructed on a smaller area of one Sq. puru~a and optional fires (klimya agni) were constructed on a bigger area of 7Y2 sq. puru~as or more, each having minimum of five ·layers of bricks. The perpetual fires had twenty-one bricks and optional fires had two hundred bricks in each layer in the first construction with other restrictions. For optional 103 fire altars, the•. whole family• of the organizer had1 to reside by the construction site of optional fire altar, for which another class of structure known as mahiivedf and other related vedfs were made. However, a summary of these types of altars with ground I shapes are grouped below: (a) Perpetual fire altars (area coverage: 1 sq. puru~a): iihavanfya (square), giirhapatya (circle or square), da~ii'Jiigni (semicircle). (b) Optional fire altars (area coverage: 7-~ square puru~a): Caturasrasyenacit (hawk bird with square body, sq uare wmgs, square tail), kankacit and alajacit (bird, with curbed wings and tail), prauga (triangle), ubhayata prauga (rhombus), rathacakracit (circle), drafJacit (trough), smasiinacit, (isosceles trapezium), kurmacit (tortoise) etc. (c) Vedis: mahiivedf or saumikf vedf (isosceles trapezium), sautramani vedf' (isosceles trapezium, and also one-third of the mahiivecff); pait~kr vedf (isosceles trapezium or a square, area one-third of sautramani vedl), pragvan:zsa . (rectangle). One can guess the natUre of knowledge, which could originate from such altar constructions. However, the Sulba-Sutras of Baudhayana, Apastamba and other schools have summarized this knowledge as available from the SaiT).hitas and Brahmaqas. Both Baudhayana and Apastamba belonged to different schools but followed a similar pattern, which also suggests that these schools inherited the knowledge from older . ~ , schools. While giving details, the Sulba-Sutras use the word vijfziiyate (known as per traditions), vedervijfziiyate (known as per Vedic tra~ition) etc. very often. A summary of this knowledge \\fill be of great interest. Baudhayana gives various units of linear measurements viz., 1 pradeia = 12 angulas; 1 pada = 15 ang; i~a = 188 ang; 1 a~a = 104 ang; 1yuga = 86 ang; 1 janu = 32 ang; 1 samyii = 36 ang; 1 biihu = 36 ang; 1 prakarma = 2 padas; 1 aratni = 2 pradesas; 1 puru.$a = 5 aratnis; 1 vyiiyama = 4 aratnis; 1 ang = 34 tilas = % inch (approx.). 104 Knowledge of rational numbers like 1, 2, 3 .. ~ 10, 11 ... 100 ... 1000, Yz, ~. lf4, Ys, 1116, 3/2, 5/12, 7Yz, 8Yz, 9Yz etc. were used in decimal word notations and their fundamental operations like addition, subtraction, multiplication and division· were carried without -:-·· ·any mistake. I Baudhayana had knowledge of square, rectangle, triangle, circle, isosceles, trapezium and various other diagra~s and transformation of one figure into another and vice­ versa. Methods of construction of square by .adding two squares or subtracting two . i squares were known. The areas of these figures were also calculated correctly. That the length, breadth and diagonal of a right triangle maintain a unique relationship, a2 + b2 = c2 (where a = length, b = breadth and c = hypotenuse) or in other words ( 1, 1, 2) (2, 1, 3) formed important triplets thereby forming an important basis for number line, and were used for construction of bricks and geometricai figures. For easy verification, Sulbakaras suggested triplets expressed in rational and irrational numbers like (3, 4, 5),' (12, 5, 13), (15, 8, 17), (7, 24~ 25), (12, 35, 37), (15, 36, 39), (1, 3, -../10), (2, 6, -../40), (1, -../10 -../11), (188, 78'lS, 203o/J), (6, 2Yz, 6Yz), (10, 41/6, 105/6) and so on. A general statement on 'Theorem of Square on Diagonal' was also enunciated thus: 'The areas (of the squares) produced separately by the length and the breadth of a rectangle together equals the area (of the square) produced by the saine diagonal'. This has been wrongly . referred to Pythagoras who was assoCiated with the triplet (3, 4, 5) and the theorem is known as the Pythagorian theorem. Indian knowledge is based on rational and irrational arithmetical facts and geometrical knowledge of transformation of area from one type to the other and its importance was perhaps correctly and perfectly realised. How the Babylonians, Egyptians, Chinese and the Greeks came upon the knowledge of ·triplets but not the general statement is equally important for an interesting study. 3 The Sulba-Sutra tradition vanished. Only a limited commentary from a late period is '. available. Whether the tradition has been lost or the elements· have been absorbed in temple architecture is still to be investigated and may be the part of other studies. I 105 Decimal scale, decimal place-value, numerical symbols and zero : 'Our numerals. and the use of zero', observes ,Sarton4 (1955), 'were invented by the Hindus and transmitted to us by th~ Arabs (hence the name Arabic numerals which we gave them)'. The study of Sachs, Neugebauer on Babylonian tablets, Kaya and Carra de Vaux on Greek sciences, Needham on Chinese sciences and study of Mayan culture have many interesting issues. The study of scholars likes Smith and Karpinski, Datta, Bag and Mukherjee5 have analysed Indian contributions, but still there is a need for a comprehensive volume. However, the salient points may be of interest: The Indians had three-tier system of word-numerals starting from the SamhiUis as follows: ·(a) Eka (1), dvi (2), tt; (3), catur (4),pafica (5), ~a( (6), sapta(7), a~{a (8),and nava (9). (b) Dasa (10), vimsati (2 x 10), trimsat (3 x 10), catviirimsat (4 x 10), paficiisat (5 x 10), ~a~thf (6 x 10), saptati (7 x 10), asiti (8 x 10) and navati (9 x 10). 2 4 (c) Eka (1), dasa (10), sata (10 ), sahasra (10\ ciyuta (10 ), niyuta (10\ prayuta 6 8 9 10 11 ( 10 ), arbuda (1 0\ nyarbuda ( 10 ), samudra ( 10 ), Madhya (1 0 ), anta (1 0 ) 12 andpariirdha (10 ). The names and their order have been agreed upon by almost all the authorities for (a) and (b), whereas there is a variation in (c) where mostly one or two terms have been I added later. The numbers below 100 were expressed with the help of (a) and (b) sometimes following additive or substractive principles e.g., trayodasa (3 + 10 = 13), unavimsati (20- 1 = 19), while for numbers above hundred, groups (a), (b) and (c) were used.